384
7. Physically Insignificant Fast Waves
Recalling that P = pRT,
cPsPs
Ps 1
1
o
g
pgzdz = -
+ -
g 0
RTda,
and the total vertically integrated energy per unit area is
& = cPsPs + Ps t (u .u+ CpT) da.
g
g 10 2
The total time derivatives in the momentum and thermodynamic equations must
be written in flux form in order to obtain a conservation law goveming &. For any
scalar y,
1
00
PSdi = Psfit + PsU ' Vay + psa 8a + y
dy
8y
. 8y
[8 ps
at + Va' (Psu) + 8a (psa)
8
. ]
8
8
.
= 8t (Psy) + Va' (Psyu) + 8a (psya),
(7.124)
where the quantity in square brackets is zero by the pressure-tendency equation
(7.104) . Adding Psc p times the thermodynamic equation (7.103) to the dot product of PsU and the momentum equation (7.102) and using (7.124), one obtains
8E
8
.
RTw
fit + Va ' (Eu) + 8a (Ea) + u- PsVacP + u - RTVaPs - -a- = 0, (7.125)
where E = Ps(u · u/2 + cpT) . Defining
RTw
F = -cPVa . (Psu) + u . RTV a Ps - - - ,
(7.126)
o
(7.125) may be expressed as
8E
-
8
.
+ Va' [(E + PscP)u] + -(Ea) + F = O.
(7.127)
8t
Ba
The forcing F may be written as the vertical divergence of a flux as folIows.
Substituting for w using (7.107),
F = -cPVa' (Psu) + - RT1a
a 0
Va ' (Psu)dä,
and then substituting for RT[a from the hydrostatic equation,
F = -cPVa . (Psu) - - 8cP 1 Va . (Psu) dä
8a 0
a
= - 8: [cP 1
a
Va ' (Psu)dä J.